Hybrid Systems and Hybrid Automata

1. Hybrid Systems and Hybrid Automata Lecture 18 (03/21/02)

2. What is a Hybrid System? • Dynamic systems that require more than one modeling language to characterize their dynamics • Provide a mathematical framework for analyzing systems with interacting discrete and continuous dynamics • Capture the coupling between digital computations and analog physical plant and environment (another way: interaction between time-driven signals (synchronous) and event-driven signals (asynchronous) • Continuous Dynamics: mechanical, fluid, thermal systems, linear circuits, chemical reactions • Discrete dynamics: collisions, switches in circuits, valves and pumps

3. + control algorithm actuators plant - Tsample sensors embedded controller physical system Hybrid Models of Physical Systems • Why Hybrid Models? • Proliferation of Embedded Systems • Simplify Behavior analysis of complex non linear systems Motivation(s): Need to accurately describe dynamic behavior Monitoring & Diagnosis Design, Control signal domain D/A energy domain A/D

4. Practical Applications • Manufacturing systems: • part processed by machine only after its arrival at the machine (triggered, event-driven process) • Process within machine can be described by time-driven dynamics In the past, handled separately: event-driven: automata or Petri nets; time-driven: differential or difference equations But this is not good for high performance analysis and optimization – becomes even more difficult when tight coupling exists between the two forms

5. Other Practical Applications • Automotive control – electronic fuel ignition system with embedded controller; performance parameters: reduce gas consumption, reduce emissions while maintaining performance – requires tight integration of continuous and discrete time processes (in the past discrete time behavior, e.g., 4 stroke cycle was reduced to continuous time behavior: models less accurate and computationally more complex) other examples: anti-lock system of brakes, etc.

6. Third Practical Example • Interaction of discrete planning algorithms and continuous processes or plant, e.g., spacecraft systems on deep space missions: require them to be autonomous, intelligent, highly reliable Task: design sequential supervisory controllers for continuous systems Hierarchical organization may help maintain autonomy • Initially deal with simpler examples: bouncing ball, thermostat systems, diode circuits

7. Modeling Physical Systems Continuous behavior governed by - Conservation of energy - Continuity of power Discontinuous changes artifacts of - Embedded control - Simplification of complex nonlinear system behavior Have to handle - Discrete changes inmodel topology - Initial value problem

8. Supervisory Controller Environment Decision Maker Switching Signal u1 Controller 1 u2 Controller 2 y u Plant • • • um Controller m

9. Example: Bouncing Ball ball position – x1 ball velocity – x2 acceleration – g coefficient of restitution– c  [0,1] x1 > 0  continuous flow governed by differential equation when transition condition satisfied  discretejump occurs Behavior is zeno, i.e.,infinitenumber of bounces occur in finite time interval

10. Example: Thermostat Thermostat (controller) turns on radiator between 68 & 70 degrees and turns off the radiator between 80 and 82 degrees. Result: non deterministic system – for a given initial condition there are a whole family of different executions.

11. Example: Automatic Gear Box Control Lateral position x1and velocity x2 Control signals (i) gear -- v {1,2,3,4}, (ii) throttle -- u  [-1,1] Question: Optimal Control Strategy in going from point a to b. (given gear efficiencies)

12. Classification of Hybrid Behavior • Continuous systems with phased operation • Bouncing balls • Diode circuits • Walking robots • Continuous systems controlled by discrete inputs • Thermostats • Circuits with switches • Processes with valves and pumps • Control modes • Aircraft autopilot modes – maintain altitude, maintain airspeed, maintain angle of attack, take-off, landing • Coordination processes • Multi-agent systems

13. Hybrid Automata • Hybrid automata is a 6-tuple H = (V, X, f, Init, Inv, Jump) V  I – set of discrete modes X Rn – real-valued variables, often the state vector f : V x Rn  Rn -- vector field Init  V x Rn -- defines initial state of H (v,Z) Inv  V x Rn -- invariant condition Jump: V x Rn P(V x Rn) – jump condition; what transitions from one discrete mode to another are possible, and what value should be assigned to state vector after the jump (reset condition) (v,z) V x Rn (z is an evaluation of x) stateof H

14. Hybrid Automata: Graphical Representation H  directed graph (V,E), vertices V, and edges E E = {(v,v’)  V x V :  z,z’ Rn, (v’ ,z’)  Jump(v,z)] Init(v) = {z  Rn : (v,z)  Init} Inv(v) = {z  Rn : (v,z)  Inv} G(e) = {z  Rn :z’  Rn(v,z), (v’ ,z’) Jump(v,z)} J(e,z) = {z’  Rn : (v’ ,z’) Jump(v,z)}

15. Hybrid Time Trajectory • Interval Point Paradigm Two key issues: 1. When do jumps occur, and how to model the transition ? 2. When system re-enters a continuous mode, what is the initial state vector ?

16. Inherently continuous Discontinuities attributed to modeling abstractions parameter abstraction time scale abstraction Implement discontinuities as transitions in continuous behavior systematic principles compositional modeling Dynamic Physical Systems

17. Abstraction Semantics • Parameter Abstraction • abstracts away complex non linear behaviors • intermediate modes mythical • switching model uses a posteriori state values • Time Scale Abstraction • collapses behavior in small intervals to point in time (pinnacle) • switching model uses a priori state values Our Goal: systematic model building to facilitate building Hybrid Automata for real-time analysis

18. Example: Diode-Inductor Circuit Diode-Inductor Circuit Mode Switching Switch closed: inductor charges Switch open: IL=0 Diode comes on No parasitic capacitance or resistance Sequence of instantaneous changes

19. Simulation Result • Freewheeling Diode

20. Parameter Abstractions • Principle of Invariance of State • Switching transition for parameter abstraction depends on a posteriori state vector value Lemma: Any vector that represents the state of a linear physical system is invariant across mode changes. Proof: based on converting any state vector to particular state vector involving energy variables. (Mosterman, Biswas, and Sztipanovits, “A hybrid modeling and verification paradigm for embedded control systems,” Control Engineering Practice, vol. 2, pp. 127-142, 1998.) Conjecture: This may be extended to nonlinear systems provided an inverse mapping can be computed uniquely.

21. Time Scale Abstraction • Perfect Elastic Collision • elasticity effects condensed to a point in time • conservation of state • conservation of energy • Collision Chain • energy state changes

22. Time Scale Abstraction • State Vector change governed by the principle of Conservation of State. • Mode change from interval to point to interval. Point is called a pinnacle. E.g., colliding bodies Newton’s Collision rule: v2+- v1+=  (v2 - v1 ) ;  - coefficient of restitution and equate Forces: m1 (v1+- v1 ) = m2 (v2+ - v2 )

23. Hybrid Systems: Issues and Challenges • Building Hybrid Models of Complex Systems • Systematic introduction of abstraction phenomena • Composing hybrid automata • Design of Hybrid Systems • Verification and Validation of Hybrid Trajectories • Monitoring and Control of Hybrid Systems • Issues of switching transients • Fault Detection and Isolation • Combining discrete-event and continuous paradigms • Fault Adaptive Control • Fault detection, isolation, study of consequences, controller selection, transient management